" In a database of with 107 tables, only 10 had any type of relationship defined between them."
That does not neccesarily mean the design is bad though, sometimes these things are done intentionally because adding a relation would adversely affect performance. What I'm really saying is that you shouldn't force yourself to do everything 'properly', because a 'properly' designed database sometimes just doesn't cut it.
"I'm in the process of resolving these issues, however management is insistent that the site remains operational as well new development be pushed forward. "
Ah management... gotta love those guys...
One thing is ofcourse clear as a bell, you can create an offline copy of the whole thing and re-program it to fit the new database design, but at some point in time you will have to shut the whole thing down for a couple of minutes to copy the live data to the new database design and switch the scripts over.
Personally I would not re-vamp the frontend at all, because it will draw more attention to it than before, and if the new site fails (which it will, Murphy says so) you'll lose face when you have to switch back to the old site.
That reminds me, ofcourse you're going to create backups of the old data, and you'll write a tool to migrate the data from the old database to the new, but it may be worth the effort to also create a tool to convert from the new database back to the old one, in case the new design collapses after a few days.